Computational Geosciences (2018) 22:711–723https://doi.org/10.1007/s10596-018-9721-0

ORIGINAL PAPER

Analytical solution of polymer slug injection with viscous fingering

S. A. Abdul Hamid1 · A. H. Muggeridge1

Received: 12 June 2017 / Accepted: 5 January 2018 / Published online: 8 February 201 8© The Author(s) 2018. This article is an open access publication

AbstractWe present an analytical solution to estimate the minimum polymer slug size needed to ensure that viscous fingering ofchase water does not cause its breakdown during secondary oil recovery. Polymer flooding is typically used to improveoil recovery from more viscous oil reservoirs. The polymer is injected as a slug followed by chase water to reduce costs;however, the water is less viscous than the oil. This can result in miscible viscous fingering of the water into the polymer,breaking down the slug and reducing recovery. The solution assumes that the average effect of fingering can be representedby the empirical Todd and Longstaff model. The analytical calculation of minimum slug size is compared against numericalsolutions using the Todd and Longstaff model as well as high resolution first contact miscible simulation of the fingering.The ability to rapidly determine the minimum polymer slug size is potentially very useful during enhanced oil recovery(EOR) screening studies.

Keywords Polymer slug · Viscous fingering · Enhanced oil recovery · Method of characteristics

1 Introduction

Polymer flooding is the most widely used chemicalenhanced oil recovery (EOR) technique in the world, withmore incremental oil recovery attributed to this methodthan all other types of chemical EOR combined [21]. It istypically used in more viscous oil reservoirs (1 < μo <

100cp) as mixing polymer into the injected water increasesthe viscosity of the aqueous phase and in turn reduces itsmobility relative to the oil. It is thus sometimes referredto as augmented waterflooding (e.g., [2]) or water-basedflooding, as the physical properties of the injected water aremodified by adding the polymer. It is particularly relevantand attractive today, because oil companies are increasinglylooking to develop more viscous oil fields as the fields withlighter crudes become mature.

Injecting polymer solution rather than water results ina higher shock front saturation compared to an ordinary

� S. A. Abdul Hamidsaa1314@imperial.ac.uk

A. H. Muggeridgea.muggeridge@imperial.ac.uk

1 Department of Earth Science and Engineering,Imperial College London, London, SW7 2BP, UK

waterflood, reducing watercut until the polymer front breaksthrough, as well as improving the overall macroscopicsweep efficiency in heterogeneous reservoirs [9]. It can alsohelp stabilize displacements for which the oil-water shockfront mobility ratio is greater than 1 [5], thus reducing orpreventing the degree of viscous fingering. An example ofthis is shown in Fig. 1. Here, the oil is 50 times moreviscous than water, so the injected water front is unstableand immiscible fingering between the injected water andresident oil is observed (see Fig. 1a). Dissolving polymerin the injected water, so that the viscosity of the aqueousphase matches that of the oil, makes the polymer front stablealthough the leading shock front between the connate waterbank and the oil may still be unstable (Fig. 1b). Figure 1cshows the average saturation profiles between the injectionand production wells for the two cases.

As for other EOR techniques, the additional costsassociated with polymer injection are typically higher thanwaterflooding. The main extra cost is the polymer itself,incurred continuously as operating expenditure (OPEX).Sheng et al. [24] reported that the average cost of polymer(excluding related costs such as processing cost) is aroundUSD4.00 per barrel of incremental oil achieved, which canbe a substantial addition to the oil lifting cost.

Costs can be reduced by injecting a fixed volume (orslug) of polymer solution, followed by water. However, carehas to be taken to ensure that the polymer slug is sufficiently

712 Comput Geosci (2018) 22:711–723

Fig. 1 Illustration of thestabilization of viscous fingeringby polymer injection in a linedrive case with an oil-waterviscosity ratio of 50 and apolymer-oil viscosity ratio of 1.a Saturation distribution seen inwater injection. b Saturationdistribution seen in polymerinjection. c Average watersaturation between the injectorand the producer from (a), (b). dOil recovery curves. All shownat 0.1 PVI. All the parametersused in this simulation are as perTable 1

(a) (b)

xd

Sw 0.5

1(c)

Water InjectionPolymer Injection

td

0 0.5 1 0 0.5 1 1.5

Np 0.3

0.6(d)

large that it maintains its integrity from the injection to theproduction well.

Various factors including adsorption and viscous finger-ing can potentially destroy the slug. Adsorption means thatpolymer is lost progressively from the slug as it movesthrough the reservoir. If the slug is too small then all thepolymer may be adsorbed before it reaches the productionwell. Viscous fingering occurs at the trailing edge of thepolymer slug because the chase water is less viscous andthus more mobile than the polymer solution. Miscible vis-cous fingers of water tend to form and grow into the polymerslug. If these fingers reach the leading edge of the slug thenthe slug integrity is destroyed and the benefits of polymerinjection are lost (Fig. 2). A number of authors have pre-sented analytical solutions that can be used to estimate theimpact of adsorption on a polymer slug including [3, 20, 22,24], and [8], but to date no-one has derived a solution todescribe the impact of viscous fingering.

Very high resolution simulations are required to properlycapture the growth and development of viscous fingers(e.g., [10, 13]). Using the required number of grid cellsis impractical, if not impossible, especially in field-scalestudies. This has driven the development of empiricalfingering models which capture the average behavior of a

fingered front. These were originally derived for applicationin miscible gas injection ([14, 16, 25]) but Bondor et al. [7]proposed that the Todd and Longstaff model [25] could alsobe used to describe the fingering of water into the rear ofa polymer slug. One drawback of these fingering models,however, is that the fitting parameters in their formulationmay need to be calibrated by comparison with detailedsimulation.

Using such empirical models to describe fingering hasenabled various authors to subsequently derive analyticalsolutions to describe various miscible gas injection pro-cesses. For example, Blunt and Christie [6] and Juanesand Blunt [15] have shown how it is possible to predictthe behavior of water alternating gas (WAG) displacementsusing the Todd and Longstaff model. Such analytical mod-els enable rapid estimation of the best ratio of water to gasto inject to maximize recovery as well as providing a meansfor validating numerical models.

This paper presents an analytical expression to estimatethe minimum polymer slug size needed to maintain theeffectiveness of the polymer flood in the event of viscousfingering of the chase water case. This is obtained froma new semi-analytical solution that describes the averageeffects of the fingering of chase water into a polymer

Fig. 2 Illustration of the chasewater fingering into a polymerslug and destroying the slugintegrity. This reduces ordestroys the benefits of polymerinjection. aMap of polymerconcentration. bMap of waterphase saturation. c Averagewater saturation between theinjector and the producer from(a), (b). d Oil recovery curves.All shown at 0.5 PVI. All theparameters used in thissimulation are as per Table 1

(a) Cp (b) Sw

xd

0 0.5 1

Sw

/ C

p

0.5

1(c)

Cp

Sw

td

0 0.5 1 1.5

Np 0.3

0.6(d)

Without fingeringWith fingering

Comput Geosci (2018) 22:711–723 713

slug in 1D. We first briefly review the analytical solutionof continuous polymer injection, following which thederivation describing the dynamics of a stable chase waterfront position as a function of time is made. These solutionsare then combined and extended to capture the effect ofmiscible viscous fingering of the chase water into thepolymer slug. This is achieved by assuming that the averageeffect of fingering can be represented by the empirical Toddand Longstaff model. Finally, the validity of the analyticaltechnique is demonstrated by comparing it against itsnumerical equivalent as well as high resolution first contactmiscibility simulation.

2 Analytical solution for continuous polymerinjection

Let us first consider continuous polymer injection into ahomogeneous 1D model so that we can understand the keyfeatures of the displacement. We assume that the system isincompressible, there is no adsorption of polymer onto therock, and physical diffusion and dispersion are negligible.We define a normalized concentration of polymer solutionin the water phase,

Cp = cp

cp,inj(1)

where cp is the polymer concentration in mass of polymerper volume of water and cp,inj is the injected concentrationof polymer in the same units. Using this definitionμw(Cp = 0) = μw and μw(Cp = 1) = μp, where μw isthe water viscosity and μp is the polymer viscosity at theinjection well.

As shown in Appendix A, the conservation of aqueousphase saturation Sw and Cp can be expressed as twohyperbolic, first order equations

∂Sw

∂td+ ∂Fw

∂xd

= 0 (2)

∂CpSw

∂td+ ∂fpFw

∂xd

= 0 (3)

where Fw is the fractional flow of the water phase and fp isthe fractional flow of the polymer component in the waterphase.

If we assume the effects of gravity and capillarity arenegligible, the fractional flow of the aqueous phase Fw isgiven by

Fw = Qw

Q= 1

1 + μw

krw

kro

μo

(4)

and the fractional flow of the polymer component in thewater phase is given by

fp = Qp

Qw

(5)

where Qw is the flow rate of the water phase, Qp is the flowrate of the polymer component in the water phase, krw is therelative permeability of water, μw is the water viscosity, krois the relative permeability to oil and μo is the viscosity ofoil. In this paper, we assume krw and kro curves depend onlyon water phase saturation (shown in Fig. 6) and any changesin the fractional flow function (4) are due only to changesin μw when the polymer is added to the water phase. Inthe absence of viscous fingering, fp = Cp as the polymersolution and the chase water are first contact miscible. tdand xd are dimensionless time and dimensionless distancein the direction of flow, respectively. They are defined bythe following expressions

xd = x

L(6)

td = Qt

φAL(7)

where L is the distance between the injection and produc-tion wells, A is the cross-sectional area (perpendicular toflow), φ is the porosity and Q is the injection rate.

For a line drive in which polymer is injected continuouslyinto a reservoir containing oil and immobile connate water(at saturation Swc), the initial conditions are

Sw = Swc, Cp = 0; for all xd

If the injected water phase’s viscosity remains constant(i.e., continuous water injection without polymer orcontinuous secondary polymer solution injection), we thenhave a Riemann problem in the half-plane which can besolved by using the method of characteristics (MOC), withxd/td as the self-similar variable. Following Pope [19], thesolution to this problem can be obtained graphically usingtwo fractional flow curves, for water-oil (Fw(Sw, Cp = 0))and polymer-oil (Fw(Sw, Cp = 1)) as shown in Fig. 3a.Note that we use Cp = 1 here for simplicity. The followingsolutions are valid for any values of 0 ≤ Cp ≤ 1 as longas the Fw curves are correct for the selected concentrationvalues.

The resulting solution for Sw against xd is shownin Fig. 3b. Unlike the Buckley-Leverett solution forwaterflooding, the profile of Sw against xd is characterizedby the existence of two shocks and a spreading wave. Thefirst shock with saturation S1 exists due to the formation of aconnate water bank (Cp = 0) that has been displaced by themore viscous polymer. This is followed by another shock,at which Cp increases from 0 to 1, and water saturationincreases from S1 to S2. S2 is found by the tangent tothe polymer fractional flow curve that goes through origin

714 Comput Geosci (2018) 22:711–723

Sw

F w

0,0 Swc

S1

S2

1-Sor

v1

v2

Cp=0

Cp=1

(a)

xd

S w

x1x2

Swc

S1

S2

1-Sor

Cp=1

Cp=0

(b)

Fig. 3 1D solution for continuous secondary polymer injection. aFractional flow curves for water-oil and polymer-oil that are used toconstruct the solution shown in (b). b Secondary polymer floodingis different from waterflooding (in dashed green line) as there aretwo shocks, one at the beginning of the connate water bank withsaturation S1 and another at the beginning of the polymer solution withsaturation S2

as illustrated in Fig. 3a, while the “jump” from waterto polymer curves (S1 to S2) must satisfy the followingcondition which imposes the conservation of mass at thevicinity of the shock

Fw (S2) − Fw (S1)

S2 − S1= Fw (S2)

S2(8)

3 Analytical solution for polymer sluginjection

Having understood the key features of continuous polymerinjection into an oil reservoir with no mobile water initially,we now examine how this is modified when we inject aslug of polymer followed by chase water. This analyticalsolution is complicated by the fact that there is a thirddiscontinuity that occurs between the chase water and thetrailing edge of the polymer slug, in addition to the twoshocks described in the previous section. As noted in theintroduction, analytical solutions to this problem have beenpresented by Bedrikovetsky [3], Ribeiro et al. [22], de Paulaand Pires [20], Borazjani et al. [8], and Vicente et al.[26]. However, these solutions are more mathematicallycomplex than those presented here; moreover, they focusedon assessing the effects of adsorption and did not considerthe effect of viscous instability.

3.1 No fingering

We first consider the case when there is no fingering ofthe chase water into the polymer slug. Polymer injection

is stopped at time td,slug and followed by chase water. Theinjection rate of the chase water is the same as the injectionrate used to inject the polymer solution.

In the case of chase water injection, a further discon-tinuity forms at the trailing edge of the polymer slug, inaddition to the shock at the front of the connate water bankand the shock at the leading edge of the polymer slug.This continuity does not travel at constant speed, unlike thefirst two shocks. This was first analyzed by Bedrikovet-sky [3] using Green’s theorem; however, we shall explainthe dynamics of this discontinuity more simply here usinga geometric construction in the graph of fractional flowagainst water saturation. This will enable us to describe thelate time behavior of the discontinuity and subsequentlyderive an analytical expression for estimating the minimumslug size in the presence of viscous fingering in the nextsubsection.

The boundary conditions corresponding to the injectionof the chase water are

Sw = 1, Cp = 1; 0 < td < td,slug

Sw = 1, Cp = 0; td > td,slug

The discontinuity between the chase water and the polymerslug is found at distance x3 from the inlet. The watersaturation immediately downstream of the discontinuity isS3 and the water saturation immediately upstream is S4.

Let us now find x3 and saturation S3. First, select a valueS3 on the polymer-oil fractional flow curve rememberingthat S2 < S3 < 1 − Sor. We then draw a line tangentto this fractional flow curve with a gradient of v3 =dFw (S3) /dSw. This line passes through the y-axis andx-axis at points A and B respectively as illustrated in Fig. 4a.

Now consider Fig. 4b. The area under the curverepresenting the slug is given as the sum of A1 and A2.For the polymer volume to be conserved (assuming noadsorption), this area must equal the injected volume oftd,slug. Hence, we have

td,slug = S2(v2td − x3) +∫ S3

S2

xd dSw − (S3 − S2)x3

where v2 is the characteristic velocity of the leading edge ofthe polymer slug.

Since for xd ≥ x3, xd = (dFw(Sw,Cp=1))dSw

td , we then have

td,slug = S2v2td − S2x3 + (F3 − F2)td − (S3 − S2)x3

where F2 = Fw(Sw = S2, Cp = 1) and F3 = Fw(Sw =S3, Cp = 1).

From Buckley-Leverett theory and the Welge construc-tion (Fig. 4a), F2 = S2v2. Subsequently,

td,slug = S2v2td + F3td − S2v2td − S3x3

td,slug = F3td − S3x3 (9)

Comput Geosci (2018) 22:711–723 715

Since x3 = v3td , (6) can be rearranged to give

td,slug

td= F3 − S3v3 = A (10)

x3 = v3td = Atd

B(11)

Note that when td = td,slug, x3 = 0 and so F3 = 1. Thisindicates that the trailing edge of the slug is at the injectionwell and the water saturation at the injection well is 1, asexpected. As td increases, F3 − S3v3 decreases reflectinga decreasing value of S3 which causes F3 to decrease andv3 to increase (remember that S2 the leading shock frontsaturation of the polymer slug remains constant). To find thelimiting value of S3 (the saturation at the trailing edge of thepolymer slug) and v3, we let td go to infinity in Eq. 10. Thisgives v3 = F3/S3. This can only be true if S3 has decreasedto S2 so the trailing edge of the polymer slug is traveling atthe same speed as the leading front, i.e.,

v3 = F3

S3= F2

S2= v2 (12)

This means that at late times, we have a rectangularsaturation profile for the slug. The length of the slug, x2−x3,is simply given by

x2 − x3 = td,slug

S2(13)

At the chase water front, the value of S4 can be deter-mined graphically from S3 at any given time by performinga jump from polymer-oil curve to water-oil curve in Fig. 4asimilar to Eq. 8

Fw(S4) − Fw(S3)

S4 − S3= Fw(S4)

S4(14)

Sw

F w

0,0

S2

S3

S4

v3 v4

A

-B

(a)

xd

S w

x1x2x3

S2

S3

S4

Cp=0 A1

A2

(b)

Fig. 4 1D solution for polymer slug injection. a Fractional flow curvesfor water-oil and polymer-oil. The blue dashed line with a gradient ofv3 that goes through points A and B can be used to calculate x3 using(11). The change in saturation from S3 to S4 can be determined bytaking a jump between the two curves using a line that goes through theorigin, S3 and S4. b The position of slug trailing edge x3 is determinedby considering the area A1 + A2 which should be equal to td,slug

3.2 Empirical model of miscible viscous fingering

The discussion thus far assumes that the interface betweenchase water and slug tail is stable without any occurrenceof viscous instability. In reality, the interface is unstable andviscous fingering of the chase water into the polymer slug isexpected as the chase water is less viscous than the polymerslug and the water and polymer solution are first contactmiscible. In principle, high resolution simulation is requiredto model the evolution of each of the fingers; however, asnoted previously, Bondor et al. [7] proposed using the Toddand Longstaff model to describe the average behavior of thewater and polymer components when this fingering occurs.This is now the standard way of representing the effects ofviscous fingering in commercial simulations because (a) itcan be implemented relatively easily in the existing blackoil simulator framework and (b) it provides a fast way ofapproximating the likely impact of viscous fingering of thechase water on a polymer slug. Bondor et al. [7] proposedthat fingering of the chase water into the polymer slug couldbe treated as the fractional flow of two components of thewater phase: the chase water and the polymer slug. Theeffective polymer slug and chase water viscosities in eachgrid block, μp,eff and μw,e, should be calculated using:

μp,eff = μm(Cp)ωμ(1−ω)p (15)

μw,e = μm(Cp)ωμ(1−ω)w (16)

where μp is the viscosity of the polymerized water atmaximum polymer concentration (Cp = 1), μw is thepure water viscosity, and μm(Cp) is the viscosity of amixture of pure water and polymer solution as a functionof the polymer concentration. ω is a mixing parameterthat can vary between 0 (no fingering) and 1 (completemixing) but is typically set to 0.67 when there is fingeringmainly based on the calibration made against experimentalresults made earlier by Blackwell et al. [4]. In general, thisneeds to be calibrated by comparison with detailed fingeringsimulations. The effective aqueous phase viscosity, μw,eff,used in the 2-phase black oil model is then calculated using

1

μw,eff= 1 − Cp

μw,e

+ Cp

μp,eff(17)

3.3 Analytical solution with fingering

As noted above, existing analytical solutions describing thedynamics of injecting a polymer slug followed by chasewater ignore the impact of the chase water fingering intothe trailing edge of the polymer slug. We now develop anapproximate analytical solution that captures this effect andshow that the resulting solution can be used to estimate theminimum polymer slug size needed to ensure that it is not

716 Comput Geosci (2018) 22:711–723

destroyed by fingering before it reaches the injection well.We achieve this by using the Todd and Longstaff modelto represent the average effects of the fingering. To incor-porate miscible fingering between the polymer slug and thechase water using the Todd and Longstaff model, we takethe mass balance Eq. 3 and replace the fractional flow fp

with

fp = 1

1 + μp,effμw,e

1−Cp

Cp

= 1

1 + μp

μw

(1−ω) 1−Cp

Cp

(18)

using Eqs. 15–16. An analogous formulation for the water-oil-solvent system was used by Blunt and Christie [6]and more recently by Juanes and Blunt [14] to derive ananalytical solution for WAG injection, although in WAGinjection the fingering takes place at the front betweenthe solvent (gas) and the displaced oil. They found exactsolutions by simultaneously solving the conservation Eqs. 2and 3.

From Eqs. 12–13, we assume that the water phasesaturation in the polymer slug is constant and given bythe saturation at the leading edge of the polymer slug. Wefurther justify this assumption with the observation that thissaturation is typically high and the change in saturationacross the slug, from trailing to leading edge, is relativelysmall. This assumption means that, we can reduce (3) to

S2∂Cp

∂td+ F2

∂fp

∂xd

= 0

S2∂Cp

∂td+ F2

dfp

dCp

∂Cp

∂xd

= 0 (19)

In this case, we can obtain the dimensionless position of agiven mean concentration of polymer Cp, from the injectionwell, using

xd,Cp∼= v2(td − td,slug)

dfp

dCp

∣∣∣∣Cp

(20)

For example, by using ω = 0.67 in Eq. 20 allowsvisualization of the spreading wave at the back of the slug asshown in Fig. 5. The front of the spreading wave (Cp = 1)characterizes the finger tips, which travel at a faster speedthan the slug front. Its approximated velocity is given by

vtips = v2dfp

dCp

∣∣∣∣Cp=1

(21)

Eventually, the finger tips will reach the slug front andcause the slug to break down. The time at this occurs whenthe blue line (giving the approximate characteristic speedof the leading fingers) intersects the red line (giving thecharacteristic speed of the leading edge of the polymerslug).

td

0 0.2 0.4 0.6 0.8 1

x d

0

0.2

0.4

0.6

0.8

1

Cp=1

Cp=0

x2

x3

x3 asymptote

=0.67

Fig. 5 Incorporation of Todd and Longstaff model into the slug trailingedge using Eq. 16, assuming ω = 0.67 and using a slug size of td,slug =0.5 PV. The red line shows the evolution of the leading edge of the poly-mer slug while the solid black line shows the evolution of the trailingedge using the analytical solution derived in Section 3.1. The dashedblack line shows the motion of the trailing edge if the water satura-tion in the polymer slug is constant and equal to the leading shockfront saturation. The upper blue line shows the motion of the fingertips using the Todd and Longstaff model, which indicates that the fin-gers will reach the leading edge of the polymer slug at approximatelytd = 0.69. The lower blue line shows the motion of the trailing edgeof the fingering. All data used in this calculation is from Table 1

The minimum slug size needed to ensure that the chasewater fingers only just cross the polymer slug by the timethe polymer breaks through is given by

td,slug,min = S2

F2

⎛⎜⎜⎜⎝1 − 1

dfp

dCp

∣∣∣∣Cp=1

⎞⎟⎟⎟⎠ (22)

Table 1 Summary of data used in this study

Parameters Values

Fluid viscosities

Oil μo = 50

Water μw = 1

Polymer μp = 50

Water-polymer mixture viscosity μm = (1−Cp

μ1/4w

+ Cp

μ1/4p

)−4

Relative permeabilities

Oil ( 1−Sw−Sor1−Swc−Sor

)2, Sor = 0.2

Water, polymer ( Sw−Swc1−Swc−Sor

)2, Swc = 0.2

Reservoir initial state Sw = 0.2, So = 0.8

Injection state

0 < td < td,slug Sw = 1, Cp = 1

td > td,slug Sw = 1, Cp = 0

Grid blocks

MRST (1D model) 1000 × 1

ECLIPSE (1D model) 10000 × 1

FCM (2D model) 300 × 150

(aspect ratio=1:0.5)

Comput Geosci (2018) 22:711–723 717

Sw

0 0.2 0.4 0.6 0.8 1

k r

0

0.5

1

krw

kro

Fig. 6 Relative permeabilities as defined in Table 1 used in this study

Further detail is given in Appendix B. Equation 22 can alsobe derived graphically as presented in Section 5.

4 Evaluation of the analytical approach

We now evaluate the above analytical solution by comparingits predictions with results from detailed simulation of theviscous fingering of chase water into a polymer slug andblack oil simulation in which the Todd and Longstaff modelhas been implemented in the polymer options.

4.1 Methodology

To model miscible viscous fingering into the trailing edge ofthe slug, we used a higher order, IMPES (implicit pressure,explicit saturation), finite difference simulator developed tomodel the details of viscous fingering in three component,two phase flows ([10, 11]). This was originally written formiscible gas applications but was adapted to model polymerslug injection. Its ability to predict viscous fingering inmiscible displacements has previously been validated by

xd

0 0.2 0.4 0.6 0.8 1

Sw

/ C

p

0

0.5

' = ' = Cp,max' = 0

Sw, ECLIPSE

Cp, ECLIPSE

Sw, MRST(modified)

Cp, MRST(modified)

Fig. 8 Comparison of the concentration and saturation profilesobtained with ECLIPSE and the modified MRST model, whensimulating a polymer flood using 100 grid blocks. The modificationto the Todd and Longstaff parameter reduces smearing of the leadingpolymer front

comparison with experimental results by Christie [11],Christie and Jones [12], and Al-Shuraiqi et al. [1], amongothers.

The commercial simulator ECLIPSE [23] and an open-source simulator called MATLAB Reservoir SimulationTools, MRST [17] were used for the black oil simulationsusing the Todd and Longstaff model to capture the averageeffects of fingering. An identical set of equations governsboth simulators, and we chose to use the fully implicitscheme in both these simulators throughout the study.

All simulations used the data summarized in Table 1,unless stated otherwise. All the symbols and nomenclatureare defined in Table 2. The oil and water relative perme-ability curves are illustrated in Fig. 6. Grid dimensions werechosen following a grid refinement study.

4.2 Themodified Todd and Longstaff model

Standard implementations of polymer flooding models incommercial simulators, such as ECLIPSE, include the Todd

Fig. 7 Polymer front simulationin ECLIPSE and MRST. aComparison between MRST,ECLIPSE, and analyticalsolution. b Grid refinement inECLIPSE for ω = 1. It can beseen that many more grid blocks(∼ 10000) are needed to resolvethe leading front of the polymerslug when ω = 1. Note that theconnate water bank has alreadybroken through in this case

xd

0 0.5 1

Sw

0.2

0.4

0.6

0.8

1(a) DX=100

ECLIPSE, =0ECLIPSE, =1MRST, =0MRST, =1Analytical

xd

0 0.5 1

Sw

0.2

0.4

0.6

0.8

1(b) ECLIPSE, =1

DX=100DX=1000DX=10000Analytical

718 Comput Geosci (2018) 22:711–723

Fig. 9 Comparison of theconcentration and saturationprofiles obtained with ECLIPSEand the modified MRST model,when simulating a polymerflood using 100 grid blocks. Themodification to the Todd andLongstaff parameter reducessmearing of the leading polymerfront

xd

10

Sw 0.5

1

xd

10

Cp 0.5

1

MRST (DX=1000)ECLIPSE (DX=10000)Analytical

xd

10

Sw 0.5

1

xd

10C

p 0.5

1

(a) td = 0.1 PV

(b) td = 0.6 PV

x2 x2 x1

x3 x2 x3 x2

Fig. 10 Comparison betweenFCM simulators, MRST andanalytical model with ω = 0.67,showing that the new analyticalmodel can predict thedevelopment of the viscousfingers into the trailing edge ofthe slug

xd 1C

p0

1

FCM

MRST

Analytical

xd 1

Cp

0

1

xd 1

Cp

0

1

xd 1

Cp

0

1

(a) o/ w = 10, td = 0.6 PV

(b) o/ w = 10, td = 0.7 PV

(c) o/ w = 50, td = 0.6 PV

(d) o/ w = 50, td = 0.7 PV

Comput Geosci (2018) 22:711–723 719

and Longstaff model to describe the effect of mixingbetween polymer solution and water and its effect onwater phase viscosity. This is presumably intended toallow engineers to model the fingering of chase water intopolymer as proposed by Bondor et al. [7]; however, usingω > 0 can also result in unphysical spreading of the shockfront at the leading edge of the polymer slug in the absenceof adsorption. The front at the leading edge of the polymerslug is no longer self-sharpening and considerably more gridblocks are then required to resolve it.

This effect is illustrated for a continuous polymer flood,in Fig. 7. Figure 7a shows the polymer front obtained from1D ECLIPSE and MRST simulations with the analyticalsolution for ω = 0 (no mixing between polymer and water)and ω = 1 (complete mixing). One hundred grid blockswere used in both cases. Both simulators give similar resultsin terms of numerical accuracy (first order) and in both casesthe polymer front is smeared when ω = 1. Figure 7b showsthat 10,000 grid blocks are needed in this case to obtaina shock front close to the analytical solution. Clearly, thisis impractical for field-scale simulations. One solution isto use higher-order solvers ([18, 19]), but these are morecomputationally expensive. Alsofi and Blunt [2] used asimpler method in streamline-based simulator, whereby theweighted average fractional flow is used to segregate theflow between the regions with and without polymer.

In this study, we adapted MRST using an approachcomparable to the method proposed by Alsofi and Blunt[2]. At every time step, we calculate the maximum value ofpolymer concentration, Cp,max reached in each of the gridblocks since the beginning of injection. We then modify theTodd and Longstaff mixing parameter using

ω′ = ωCp,max (23)

where ω is the input value of the Todd and Longstaff mixingparameter. This means that the mixing parameter is zero incells that have never seen polymer and reduced in cells thatsee polymer increasing (as the polymer slug advances) butis equal to the input value at the trailing edge of the polymerslug. This strategy was found to give better resolution of theleading polymer front in MRST as shown in Fig. 8.

4.3 Results—no viscous fingering

We first verify that the black oil simulators can reproducethe analytical solution derived in Section 3.1 for polymerslug injection for the case without viscous fingering.Figure 9 shows the comparison between the analyticalsolution and the predictions of the modified MRST usingω = 1 for a slug volume of 0.4 PV. Other input data isas per Table 1. A very high resolution ECLIPSE model(DX = 10000) is also shown here.

The simulators correctly capture the location of thetrailing edge x3 and the relatively small jump in saturationfrom S3 to S4 at x3. The immobile oil region, formed by thelocus of chase water front saturation S4, is also observed.

4.4 Results—viscous fingering

We now evaluate the ability of the new analytical model topredict viscous fingering of the chase water into the trailingedge of the polymer slug by comparing its predictionswith those obtained when we model the fingering explicitly.Here, we consider two cases, μp/μw, of 10 and 50.We keep μp = μo in all cases so that the potentialimpact of immiscible fingering at the slug front can beruled out. The polymer slug size is 0.5 PV. We alsocompare these two predictions with those obtained from themodified MRST using the Todd and Longstaff model withω = 0.67.

Figure 10 shows that there is very good agreement bet-ween the three approaches suggesting that the assumptionsmade in the derivation of the analytical model are appro-priate. In particular, the time when the finger tips reach thepolymer front is successfully predicted analytically.

5Minimum slug size computation

We now show how the analytical model can be used tocalculate the minimum polymer slug size graphically thatwill maintain its integrity between injection and productionwells even if there is fingering of the chase water into thetrailing edge of the slug.

td

0 0.2 0.4 0.6 0.8 1

x d

0

0.2

0.4

0.6

0.8

1

Cp=1

Cp=0

td,slug,min

x2

x3

x3 asymptote

td,slug=0.3 PV

td,slug=0.52 PV

Fig. 11 Optimal slug size estimation for polymer-water viscosity ratioof 50. The red line shows the evolution of the leading edge of thepolymer slug while the solid black line shows the evolution of thetrailing edge for td,slug = 0.3 using the analytical solution derived inSection 3.1. The upper blue line shows the motion of the correspondingfinger tips using the Todd and Longstaff model (assuming ω = 0.67)which indicates td,slug = 0.3 is too small and causes the slug to breakdown in the middle of the reservoir. The optimal slug size can bedetermined by drawing a line with a gradient of vtips given in Eq. 21that meets x2 at xd = 1. All data used in this calculation is from Table 1

720 Comput Geosci (2018) 22:711–723

(a) td,slug=0.3 PV (b) td,slug=0.52 PV

(c) td,slug=0.65 PV

td

0 0.5 1 1.5

Np

0

0.2

0.4

0.6(d)

Water Injection0.3 PV0.52 PV0.65 PV

Fig. 12 Oil recovery for different slug sizes. a A slug size of 0.3 PV istoo small as fingering of the chase results in the breakdown of the poly-mer slug before the slug reaches the production well. b The fingeringpattern seen at the same time for a 0.52 PV slug. The fingers have justreached the leading edge of the slug as the polymer front reaches the

production well. c Very little fingering seen for a slug size of 0.65 PV.d The recovery curves obtained from the 0.52 and 0.65 PV slugs arevirtually identical, whereas recovery is reduced for the 0.3 PV slug.The minimum slug size that will not be destroyed by viscous fingeringat polymer breakthrough is 0.52 PV

Figure 11 shows the evolution of the fingering for td,slug=0.3 PV on an xd − td plot. The red line shows the evolutionof the leading edge of the polymer slug while the solidblack line shows the evolution of the trailing edge usingthe analytical solution derived in Section 3.1. The dashedblack line shows the motion of the trailing edge if the watersaturation in the polymer slug is constant and equal to theleading shock front saturation. The upper blue line showsthe motion of the finger tips using the Todd and Longstaffmodel, assuming ω = 0.67. The lower blue line shows themotion of the trailing edge of the fingering. Clearly, withtd,slug = 0.3 PV, we expect the slug will start to breakdownin the middle of the reservoir at around td = 0.41 PV.

The smallest slug size to minimize the breakdown of theslug by fingering should be selected such that the fingertips reach the production well (xd = 1) at the same timeas the polymer front. This can be graphically determined bydrawing a line with a gradient of vtips given in Eq. 21 that

meets x2 at xd = 1. The green line in Fig. 11 illustratesthis. Then, td,slug,min can be directly determined from thevalue of the x-intercept. Assuming ω = 0.67, we found thattd,slug,min = 0.52 PV.

To verify this, we ran the FCM simulation for td,slug of0.3, 0.52, and 0.65 PV. The results are shown in Fig. 12.As expected, td,slug = 0.3 PV is too small and viscousfingering of the chase water causes the slug to break downin the middle of the reservoir. This is reflected in the oilrecovery plot in Fig. 12d which indicates that the polymerbreakthrough at the production well occurs rather early, ataround td = 0.5 PV. We also observe that the recoveryis less efficient as the increase in oil is very gradual postpolymer breakthrough. The recoveries obtained for td,slug =0.65 PV and td,slug = 0.52 PV are found to be almostidentical, which means that, in this case, the minimum slugsize needed to ensure fingering of the chase water does notdestroy the slug is td,slug = 0.52 PV.

Fig. 13 Optimum slug size as afunction ω and viscosity ratios.All data used in this calculationis from Table 1, using Eq. 22

100

o/ w

500.60.4

0

0.2

0.4

0.6

0.2

Slug

siz

e, t d,

slug

(PV)

10

p/ o

550

o/ w

0.4

0.6

0

0.2

100

Slug

siz

e, t d,

slug

(PV)

(a) p= o (b) =0.67

Comput Geosci (2018) 22:711–723 721

The minimum slug size needed to ensure fingering of thechase water does not completely break down the polymerslug for various values of ω and viscosity ratios calculatedusing Eq. 22 are shown in Fig. 13. We can see here theexpected trend showing that high oil viscosities or low ω

values necessitate the use of larger slug size. Generallyin polymer flooding, polymer viscosity needs to be highenough that the oil can be swept more efficiently thanwaterflooding. However, interestingly, too high a polymer-oil viscosity ratio, μp/μo, has an adverse effect instead—alarge slug size is required as the chase water tends to fingermore through the polymer slug as we can see in Fig. 13b.

Although we need to consider many parameters tocalculate the required slug size, for moderate values of ω

of say around 0.4 to 0.8, Fig. 13 suggests that td,slug,min ofaround 0.5 to 0.6 PV is a good first approximation that maybe used in the field.

6 Conclusion

We have investigated the fingering of chase water intoa polymer slug during secondary polymer flooding inthe absence of adsorption using a mixture of numericalsimulations and analytical approaches. Both detailednumerical simulations, describing the fingers explicitly,and black oil simulations using a Todd and Longstaffmodel to represent the average effects of the fingeringwere performed. We have reviewed the existing analyticalsolutions that predict both continuous polymer flooding anda slug of polymer followed by chase water in the absence offingering.

Existing solutions describing the injection of chasewater following a polymer slug assume a stable interfacebetween chase water and polymer slug. A graphical solutionbased on Welge analysis was presented in Section 3.1 andcompared successfully against numerical simulation. Weshowed analytically that at late times the water saturationthroughout the polymer slug tends to the saturation of theleading shock of the polymer slug.

We have extended this analysis to obtain an approximatemethod for predicting the growth of the fingering of chasewater into the polymer slug due to fingering, in the absenceof adsorption. This analysis provides a simple analyticalexpression that can be used to estimate the minimumpolymer slug size needed to ensure that it is not destroyed byfingering of the chase water before polymer break through.

Comparison of detailed numerical simulations of thefingering of chase water with black oil simulations usingthe Todd and Longstaff model to represent the averageeffects of the fingering have shown that a value of ω =0.67 can be used, for oil-water viscosity ratios of 10 and50. However, due to the empirical nature of the Todd

Table 2 Nomenclature

Symbol/ Definition

abbreviation

A Reservoir cross-sectional area

cp Polymer concentration in mass of polymer

per volume of water

Cp Normalized polymer concentration

cp,inj Injected polymer concentration in

mass of polymer per volume of water

Cp,max Maximum value of normalized polymer

concentration in each of the grid blocks

DX Number of grid blocks in x direction

DY Number of grid blocks in y direction

DZ Number of grid blocks in z direction

fp Fractional flow of polymer

Fw Fractional flow of aqueous phase

L Reservoir length

M Mobility ratio

Me Effective mobility

Np Oil production

PV Pore volume

Q Total injection flow rate

Qp Polymer injection flow rate

Qw Water injection flow rate

S1 Connate water bank saturation

S2 Slug front water saturation

S3 Slug trailing edge water saturation

S4 Chase water front saturation

So Oil phase saturation

Sor Irreducible oil saturation

Sw Water saturation

Swc Connate water saturation

t Time

td Dimensionless time

td,slug Dimensionless slug size

td,slug,min Minimum dimensionless slug size

v1 Connate water bank speed

v2 Slug front water speed

v3 Slug trailing edge water speed

v4 Chase water front speed

vd Characteristic speed

vtips Approximated finger tips velocity

vh Hydrocarbon phase velocity

vs Spreading wave speed

vt Total velocity

vw Aqueous phase velocity

x1 Connate water front position

x2 Slug front position

x3 Slug trailing edge position

xd Dimensionless distance in direction of flow

μm Mixture viscosity

722 Comput Geosci (2018) 22:711–723

Table 2 (continued)

Symbol/ Definition

abbreviation

μo Oil viscosity

μp Polymer viscosity

μp,eff Polymer effective viscosity

μw,e Water effective viscosity

μw,eff Aqueous effective viscosity

φ Fractional rock pore volume

ω Todd and Longstaff mixing parameter

ω′ Modified ω

and Longstaff model, this value may not work for caseswith heterogeneous reservoirs, very high viscosity ratios, orpolymer-oil viscosity ratios that are less than 1. We expectcalibration of ω is required for more general problems.When ω is provided, the optimum slug size can be rapidlydetermined using the approximate analytical solution. Suchresults can potentially be used during EOR screening orfeasibility studies, during which only order of magnitudeestimations are required and more accurate but highlyexpensive computational study may not be necessary.

We expect that the analytical solution presented can beextended to include the effects of adsorption and possibly toestimate flow behavior when polymer is injected subsequentto a waterflood.

Acknowledgements We thank Bilal Rashid for help with modifyingthe detailed viscous fingering code and advice on postprocessing. Wealso thank Schlumberger for providing the ECLIPSE software and theMalaysian Government for funding S.A. Abdul Hamid’s PhD.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.

Appendix A

In this appendix we derive the conservation equations forthe water phase and the polymer solution. Consider theconservation of mass for the water phase in one dimension

∂Sw

∂t+ ∂vw

∂x= 0. (A.1)

where we have assumed that water is incompressible. Wedefine the fractional flow Fw of the water phase as

Fw = Qw

Q= vw

vt

(A.2)

Cp

0 0.2 0.4 0.6 0.8 1

f p

0

0.5

1

=1=0.67

Fig. 14 Fractional flow fp as a function of polymer concentration Cp

for μp/μw = 50

where vt is the sum of aqueous and hydrocarbon phasevelocities, vt = vw + vh. Substituting (A.2) into (A.1), wethen have∂Sw

∂t+ ∂Fwvt

∂x= 0. (A.3)

Similarly, the conservation of the polymer component Cp inthe water phase is given by

∂CpSw

∂t+ ∂vp

∂x= 0. (A.4)

where vp is the velocity of the polymer component. We canalso define the fractional flow of polymer component in thewater phase as

fp = vp

vw

= vp

Fwvt

(A.5)

We can therefore rewrite (A.4) as

∂CpSw

∂t+ ∂fpFwvt

∂x= 0 (A.6)

In the absence of fingering then the fractional flow of thepolymer solution is simply the dimensionless concentrationof the polymer in the water phase, Cp, as the polymersolution is first contact miscible with the injected chasewater. In the presence of viscous fingering, the averagefractional flow fp can be described using the Todd-Longstaff formulation in Eq. 18. A plot of fp as a functionof Cp for μp/μw of 50 is shown in Fig. 14. Note that forω = 1, we have fp = Cp which models a fully-mixed,piston-like displacement.

Appendix B

We present here the estimation of the minimum slugrequired in order to maintain its integrity. Recall that theslug front travels at velocity v2 = F2S2 and it arrives at theproduction well when x2 = 1, hence

x2 = v2td = 1 (B.1)

Comput Geosci (2018) 22:711–723 723

With no fingering, the back of the slug will have approxi-mately the same velocity as the slug front (as discussed inSection 3.1) so v3 = v2. Hence, we have

x3 ∼= v2(td − td,slug) (B.2)

To take the fingering into account, we multiply x3 in (B.2)by dfp/dCp. The fastest wave occurs when Cp = 1, shownas the upper blue line in Fig. 5. At the production well, wehave

xd,Cp=1 ∼= v2(td − td,slug)dfp

dCp

∣∣∣∣Cp=1

= 1 (B.3)

The minimum slug size needed to ensure that the chasewater fingers only just cross the polymer slug by the timethe polymer breaks through is found by substituting (B.1)into (B.3)

td,slug,min = 1

v2

⎛⎜⎜⎜⎝

dfp

dCp

∣∣∣∣Cp=1

− 1

dfp

dCp

∣∣∣∣Cp=1

⎞⎟⎟⎟⎠ (B.4)

td,slug,min = S2

F2

⎛⎜⎜⎜⎝1 − 1

dfp

dCp

∣∣∣∣Cp=1

⎞⎟⎟⎟⎠ (B.5)

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